- Email: [email protected]
A variational calculation of nuclear binding energies
Nuclear Physics A l l 5 (1968) 516--520; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
A VARIATIONAL CALCULATION OF NUCLEAR BINDING ENERGIES V. R. P A N D H A R I P A N D E
Tata Institute of Fundamental Research, Bombay 5, India Received 20 February 1968
Abstract: We show that the interaction energy of a nuclear system interacting with a central Yukawa plus a velocity-dependent ~-function two-body interaction can be reasonably approximated as a function of the local densities. The resulting expressions bear general resemblance with the Weiszsacker semi-empirical mass law. A simple variational calculation is carried out with these expressions for the 4He, leO, 4°Ca and 56Ni nuclei and nuclear matter. The calculated binding energies and radii are in excellent agreement with the experimental data.
1. Introduction
The general tendencies of nuclear binding energies are well reproduced by the Weizsacker semi-empirical mass formula. The formula has a rather simple expression, suggested by the liquid-drop model, consisting of volume and surface terms. It is quite trivial to justify this treatment of nuclei as pieces of nuclear matter on the basis of the Thomas-Fermi method. As a matter of fact the classical self-consistent ThomasFermi calculations of Sayler and Blanchard 1) give excellent results on the gross energetics and sizes of stable nuclei. The present-day calculations of nuclear binding energies however are carried out within the framework of the Hartree-Fock theory with various two-body interactions that generally include either a non-local or a velocity-dependent term to bring about the necessary saturation. The expressions in these calculations are rather complicated and offer no simple extrapolation to the Weizsacker expression. In the present work we show that the interaction energy of a nuclear system interacting with a central Yukawa plus a velocity-dependent ~-function interaction can be reasonably approximated as a function of local densities and thus reduce the Hartree-Fock calculation to a Hartree calculation. The expressions in this calculation bear general resemblance with Weizsacker's expression. The binding energies of nuclear matter, 4He, 160, 4°Ca and 56Ni are calculated by a simple variational calculation using this expression. The calculated energies as well as radii are in excellent agreement with the experimental data. 2. The interaction energy
In the Hartree-Fock theory the many-body wave function W is approximated to a single Slater determinant and the (kU]H[~f) is minimised with respect to variations in ~P. Even with these simplifying assumptions the interaction energy can be calculated 516
NUCLEAR BINDING ENERGIES
517
as a function of the local densities only i f the spatial exchange integral is either negligible or equal to the direct integral. The latter is exact only for a 6-function interaction while the former is known to be a very poor approximation for nuclear systems. For plane waves the Yukawa range direct and exchange integrals are given by e -'/° 4~ a3 (K~K21 ~ IK1K2) = ~ , e -r/a
4re a3
(K1K21~-~a ]K2K1) = --f2
1
1 + K2a 2
where K = K1-K 2. The kinetic energies involved in the nuclear systems correspond to K~ maximum below 2 fro- 1 and the value of the range parameter a is known to be around 1.75-1 fro. For these values the exchange integral can be well approximated by the direct plus a K 2 term. The velocity-dependent interaction (V26(r) + 6(r)V 2) has momentumspace matrix dements proportional to K 2. Thus, when using such a combination of interactions the interaction energy can be evaluated by approximating the exchange integral by the direct with most of the errors involved being absorbed in the strength parameter of the velocity-dependent interaction. For example, the neutron-neutron interaction energy for the singlet spin-state Yukawa interaction in an extended system is proportional to
fro~d3Kxfrrd3Kzrrr •Jo
rr~'ldaK2"
+ J 0 daKlJo
l+K2a ----------~
In this approximation the above is given by /'KF
/~Ka*
In fig. 1 these energies are normalised to the interaction energy in the &function approximation and are plotted against Kv. It is easily seen that in the region of practical interest (KF ~ 1.4) this approximation is within 2-3 ~ whereas the estimates in the 6-function approximation are about 15 ~ higher. These errors are estimated in the limit of an extended system. However, since the only relevant quantity is the order of the kinetic energy the above should be a reasonable approximation even for the finite nuclear systems. Since the 6-function approximation itself is within about 15 ~o the use of the 6-function in the velocity-dependent interaction is justified. The wave function of a finite nucleus with N neutrons and Z protons is approximated to a product of two antisymmetric wave functions considering the neutrons and protons as non-identical particles for the sake of simplicity. 1 1 -- 4N!.I 4 ~ . ]-A(~b~(1)... $~(N))-][A($~(1)... ~kPz(Z))],
518
V. R. PANDHARIPANDE
where
¢,,
=
The single nucleon wave functions are products of special and spin functions and for even-even systems both the spin states of the special function being degenerate are either occupied or empty. The two-body interaction considered is -- r/a
Vi 2 = ( VsPs + vTpT) ~
+ ( BsPs + BTpT)(v2cs(r) + cS(r)V2),
where pS and pT are the singlet- and triplet-state projection operators. The expectation 1.0 0.9 0.8
\
Edapp.o . z \ 0.6
\\
0.5
i 0
I I
05
: 15
\
\
I 2
\
\ IX,
,
2.5
3
KF Fig. I. The plots of Eexact/E~app. (solid line) and Eapp./E6app" (dotted line) against KF.
value of the neutron-proton interaction for the Yukawa force is given by ( 3VT + vS) E f
l@~(r.)12[(°~(rp)l 2~
I, K d _-
dr.drp
?'/a
I(3vT+ vS)f,,n(rnV(r)
dr.d,,,.
Here p" and pP are local neutron and proton densities obtained by summing inside the integral. Using the above approximation the n-n and p-p interactions are similarly written as functions of local density, the error involved being absorbed in the parameters of the velocity-dependent interaction. The triplet spin-state interaction can then contribute only through the velocity-dependent term and the singletstate interaction is given by r/a
¼vsf[p"(rl)pn(r~)+pP(ri)pP(r~)] drldr2. e
In this procedure the interaction energy is summed over all the pairs at a point.
NUCLEAR BINDING ENERGIES
519
Thus the contribution due to the velocity dependent 6-function interaction is
f o2(r)I(-(r)dr, where K 2 is the weighted average of the relative kinetic energy of all pairs at r. For an extended system K 2 ,~ EK, the average kinetic energy. Since this energy is proportional to p2 most of the contribution to it is from the inside region of the nucleus. In this region the density is almost constant and hence within the approximations of the Thomas-Fermi method K 2 should be independent of r and proportional to the mean kinetic energy. Thus the velocity dependent interaction contribution is estimated as
¼Ekin(3AT + A s) f pn(r)pP(r)dr + ¼E~in ASf[pn(r)] 2d r + ¼EP,nAS f[pP(r)] 2dr. The Eldn, Eki n . and Eki p n are the nucleon, neutron and proton mean kinetic energies. The new parameters A T and A s for the velocity-dependent interaction now include the effects of the substitution of the direct integral for the exchange in evaluating the Yukawa interaction energy. In these expressions the n-p Yukawa interaction, which is the largest, is considered exactly. The approximation in calculating the velocity dependent interaction is also exact for nuclear matter. Thus the error in the interaction energy of nuclear matter with K F < 2 f m - 1 is estimated to be less than 1 ~o. If the nuclei are considered to be uniform spheres the above integrals can be split up into volume and surface terms. The kinetic energy can also be similarly expressed in the Thomas-Fermi approximation and the binding energy expression can thus be written in a form analogous to the Weizsacker expression. For quantitative calculation of the surface term knowledge of the tpt is necessary. Within these approximations H s'P' can be written as a function of r and the problem can be solved self-consistently to obtain the tpl. In the present work a simple variational calculation restricting to cases where the form of the tp~ is approximately known is carried out to see the applicability of this interaction. 3. Calculations The q~t for nuclear matter are plane waves whereas the harmonic oscillator wave functions are known to be good approximations for light nuclei. A variational calculation is carried out for nuclear matter and *He, 160, 4°Ca and S6Ni nuclei. The density being the variational parameter for the nuclear matter and that for the nuclei being the harmonic oscillator constant. Since p" = pP the expression for the interaction energy simplifies to the following:
V--~p(rl)p(r2)e--rlad r 1drz-EkinA~ p2(r)dr, where
p" = pP = p, V-- ¼(vS+ V T) and A = ¼(AS+AT).
520
V. R. PANDHARIPANDE
The results for the following values of the parameter are given in table 1. V = 805 MeV, a = 1.75 -1 fm, A = 43.5 fm a. In these calculations the integrals are numerically computed on the CDC 3600 computer. The Coulomb energy is estimated assuming uniform spherical charge distribution and the kinetic energy of the centre of mass is subtracted in calculating the Eki n • TABLE 1
The calculated and experimental values of the binding energy and equivalent radius for uniform sphere Calculated Experimental B.E.
Re
4He xeO 4°Ca SeNi
28.8 121 338 470
2.3 3.37 4.35 4.62
B.E.
Ru
28.2 128 342 485
2.08 3.14 4.5 4.9
Nuclear matter
B.E./A
Ro
B.E./ A
Ro
15.8
1.17
15.8
1.1
The experimental binding energies are from Konig et al. ~) and the radii from Hofstadter's data 8). 4. Discussion We have thus considerably simplified the Hartree-Fock calculation at the cost of a little error. We have neglected the Mayorana exchange component in the interaction because in this approximation it would just lead to a repulsive velocity dependent term. The use of a wave-function separately antisymmetric between the neutrons and protons is purely for the sake of simplicity. The expressions obtained by considering a totally antisymmetric wave function are very similar to those used. Recently Brink and Boeker 4) have carried out restricted Hartree-Fock calculations with similar approximations regarding the wave-functions. They use a central 6function plus a velocity-dependent 6-function interaction. For such an interaction of course the interaction energy can be exactly written as a function of local densities. The present results are slightly better than theirs presumably because of the range considered for the central interaction. The author wishes to thank Professor B. V. Thosar for his interest in this work and Drs. B. Banerji, Nazakat Ullah and C. S. Warke for many helpful discussions. References 1) 2) 3) 4)
R. G. Sayler and C. H. Blanchard, Phys. Rev. 131 (1963) 355 L. A. Konig, J. H. E. Mattauch and A. H. Wapstra, Nucl. Phys. 31 (1962) 18 R. Hofstadter, Ann. Rev. Nuc]. Sci. 7 (1957) 231 D. M. Brink and E. Boeker, Nucl. Phys. A91 (1967) 1
A VARIATIONAL CALCULATION OF NUCLEAR BINDING ENERGIES V. R. P A N D H A R I P A N D E
Tata Institute of Fundamental Research, Bombay 5, India Received 20 February 1968
Abstract: We show that the interaction energy of a nuclear system interacting with a central Yukawa plus a velocity-dependent ~-function two-body interaction can be reasonably approximated as a function of the local densities. The resulting expressions bear general resemblance with the Weiszsacker semi-empirical mass law. A simple variational calculation is carried out with these expressions for the 4He, leO, 4°Ca and 56Ni nuclei and nuclear matter. The calculated binding energies and radii are in excellent agreement with the experimental data.
1. Introduction
The general tendencies of nuclear binding energies are well reproduced by the Weizsacker semi-empirical mass formula. The formula has a rather simple expression, suggested by the liquid-drop model, consisting of volume and surface terms. It is quite trivial to justify this treatment of nuclei as pieces of nuclear matter on the basis of the Thomas-Fermi method. As a matter of fact the classical self-consistent ThomasFermi calculations of Sayler and Blanchard 1) give excellent results on the gross energetics and sizes of stable nuclei. The present-day calculations of nuclear binding energies however are carried out within the framework of the Hartree-Fock theory with various two-body interactions that generally include either a non-local or a velocity-dependent term to bring about the necessary saturation. The expressions in these calculations are rather complicated and offer no simple extrapolation to the Weizsacker expression. In the present work we show that the interaction energy of a nuclear system interacting with a central Yukawa plus a velocity-dependent ~-function interaction can be reasonably approximated as a function of local densities and thus reduce the Hartree-Fock calculation to a Hartree calculation. The expressions in this calculation bear general resemblance with Weizsacker's expression. The binding energies of nuclear matter, 4He, 160, 4°Ca and 56Ni are calculated by a simple variational calculation using this expression. The calculated energies as well as radii are in excellent agreement with the experimental data. 2. The interaction energy
In the Hartree-Fock theory the many-body wave function W is approximated to a single Slater determinant and the (kU]H[~f) is minimised with respect to variations in ~P. Even with these simplifying assumptions the interaction energy can be calculated 516
NUCLEAR BINDING ENERGIES
517
as a function of the local densities only i f the spatial exchange integral is either negligible or equal to the direct integral. The latter is exact only for a 6-function interaction while the former is known to be a very poor approximation for nuclear systems. For plane waves the Yukawa range direct and exchange integrals are given by e -'/° 4~ a3 (K~K21 ~ IK1K2) = ~ , e -r/a
4re a3
(K1K21~-~a ]K2K1) = --f2
1
1 + K2a 2
where K = K1-K 2. The kinetic energies involved in the nuclear systems correspond to K~ maximum below 2 fro- 1 and the value of the range parameter a is known to be around 1.75-1 fro. For these values the exchange integral can be well approximated by the direct plus a K 2 term. The velocity-dependent interaction (V26(r) + 6(r)V 2) has momentumspace matrix dements proportional to K 2. Thus, when using such a combination of interactions the interaction energy can be evaluated by approximating the exchange integral by the direct with most of the errors involved being absorbed in the strength parameter of the velocity-dependent interaction. For example, the neutron-neutron interaction energy for the singlet spin-state Yukawa interaction in an extended system is proportional to
fro~d3Kxfrrd3Kzrrr •Jo
rr~'ldaK2"
+ J 0 daKlJo
l+K2a ----------~
In this approximation the above is given by /'KF
/~Ka*
In fig. 1 these energies are normalised to the interaction energy in the &function approximation and are plotted against Kv. It is easily seen that in the region of practical interest (KF ~ 1.4) this approximation is within 2-3 ~ whereas the estimates in the 6-function approximation are about 15 ~ higher. These errors are estimated in the limit of an extended system. However, since the only relevant quantity is the order of the kinetic energy the above should be a reasonable approximation even for the finite nuclear systems. Since the 6-function approximation itself is within about 15 ~o the use of the 6-function in the velocity-dependent interaction is justified. The wave function of a finite nucleus with N neutrons and Z protons is approximated to a product of two antisymmetric wave functions considering the neutrons and protons as non-identical particles for the sake of simplicity. 1 1 -- 4N!.I 4 ~ . ]-A(~b~(1)... $~(N))-][A($~(1)... ~kPz(Z))],
518
V. R. PANDHARIPANDE
where
¢,,
=
The single nucleon wave functions are products of special and spin functions and for even-even systems both the spin states of the special function being degenerate are either occupied or empty. The two-body interaction considered is -- r/a
Vi 2 = ( VsPs + vTpT) ~
+ ( BsPs + BTpT)(v2cs(r) + cS(r)V2),
where pS and pT are the singlet- and triplet-state projection operators. The expectation 1.0 0.9 0.8
\
Edapp.o . z \ 0.6
\\
0.5
i 0
I I
05
: 15
\
\
I 2
\
\ IX,
,
2.5
3
KF Fig. I. The plots of Eexact/E~app. (solid line) and Eapp./E6app" (dotted line) against KF.
value of the neutron-proton interaction for the Yukawa force is given by ( 3VT + vS) E f
l@~(r.)12[(°~(rp)l 2~
I, K d _-
dr.drp
?'/a
I(3vT+ vS)f,,n(rnV(r)
dr.d,,,.
Here p" and pP are local neutron and proton densities obtained by summing inside the integral. Using the above approximation the n-n and p-p interactions are similarly written as functions of local density, the error involved being absorbed in the parameters of the velocity-dependent interaction. The triplet spin-state interaction can then contribute only through the velocity-dependent term and the singletstate interaction is given by r/a
¼vsf[p"(rl)pn(r~)+pP(ri)pP(r~)] drldr2. e
In this procedure the interaction energy is summed over all the pairs at a point.
NUCLEAR BINDING ENERGIES
519
Thus the contribution due to the velocity dependent 6-function interaction is
f o2(r)I(-(r)dr, where K 2 is the weighted average of the relative kinetic energy of all pairs at r. For an extended system K 2 ,~ EK, the average kinetic energy. Since this energy is proportional to p2 most of the contribution to it is from the inside region of the nucleus. In this region the density is almost constant and hence within the approximations of the Thomas-Fermi method K 2 should be independent of r and proportional to the mean kinetic energy. Thus the velocity dependent interaction contribution is estimated as
¼Ekin(3AT + A s) f pn(r)pP(r)dr + ¼E~in ASf[pn(r)] 2d r + ¼EP,nAS f[pP(r)] 2dr. The Eldn, Eki n . and Eki p n are the nucleon, neutron and proton mean kinetic energies. The new parameters A T and A s for the velocity-dependent interaction now include the effects of the substitution of the direct integral for the exchange in evaluating the Yukawa interaction energy. In these expressions the n-p Yukawa interaction, which is the largest, is considered exactly. The approximation in calculating the velocity dependent interaction is also exact for nuclear matter. Thus the error in the interaction energy of nuclear matter with K F < 2 f m - 1 is estimated to be less than 1 ~o. If the nuclei are considered to be uniform spheres the above integrals can be split up into volume and surface terms. The kinetic energy can also be similarly expressed in the Thomas-Fermi approximation and the binding energy expression can thus be written in a form analogous to the Weizsacker expression. For quantitative calculation of the surface term knowledge of the tpt is necessary. Within these approximations H s'P' can be written as a function of r and the problem can be solved self-consistently to obtain the tpl. In the present work a simple variational calculation restricting to cases where the form of the tp~ is approximately known is carried out to see the applicability of this interaction. 3. Calculations The q~t for nuclear matter are plane waves whereas the harmonic oscillator wave functions are known to be good approximations for light nuclei. A variational calculation is carried out for nuclear matter and *He, 160, 4°Ca and S6Ni nuclei. The density being the variational parameter for the nuclear matter and that for the nuclei being the harmonic oscillator constant. Since p" = pP the expression for the interaction energy simplifies to the following:
V--~p(rl)p(r2)e--rlad r 1drz-EkinA~ p2(r)dr, where
p" = pP = p, V-- ¼(vS+ V T) and A = ¼(AS+AT).
520
V. R. PANDHARIPANDE
The results for the following values of the parameter are given in table 1. V = 805 MeV, a = 1.75 -1 fm, A = 43.5 fm a. In these calculations the integrals are numerically computed on the CDC 3600 computer. The Coulomb energy is estimated assuming uniform spherical charge distribution and the kinetic energy of the centre of mass is subtracted in calculating the Eki n • TABLE 1
The calculated and experimental values of the binding energy and equivalent radius for uniform sphere Calculated Experimental B.E.
Re
4He xeO 4°Ca SeNi
28.8 121 338 470
2.3 3.37 4.35 4.62
B.E.
Ru
28.2 128 342 485
2.08 3.14 4.5 4.9
Nuclear matter
B.E./A
Ro
B.E./ A
Ro
15.8
1.17
15.8
1.1
The experimental binding energies are from Konig et al. ~) and the radii from Hofstadter's data 8). 4. Discussion We have thus considerably simplified the Hartree-Fock calculation at the cost of a little error. We have neglected the Mayorana exchange component in the interaction because in this approximation it would just lead to a repulsive velocity dependent term. The use of a wave-function separately antisymmetric between the neutrons and protons is purely for the sake of simplicity. The expressions obtained by considering a totally antisymmetric wave function are very similar to those used. Recently Brink and Boeker 4) have carried out restricted Hartree-Fock calculations with similar approximations regarding the wave-functions. They use a central 6function plus a velocity-dependent 6-function interaction. For such an interaction of course the interaction energy can be exactly written as a function of local densities. The present results are slightly better than theirs presumably because of the range considered for the central interaction. The author wishes to thank Professor B. V. Thosar for his interest in this work and Drs. B. Banerji, Nazakat Ullah and C. S. Warke for many helpful discussions. References 1) 2) 3) 4)
R. G. Sayler and C. H. Blanchard, Phys. Rev. 131 (1963) 355 L. A. Konig, J. H. E. Mattauch and A. H. Wapstra, Nucl. Phys. 31 (1962) 18 R. Hofstadter, Ann. Rev. Nuc]. Sci. 7 (1957) 231 D. M. Brink and E. Boeker, Nucl. Phys. A91 (1967) 1